Partial resolution of complex cones over Fano $\mathcal{B}$

Abstract

In our recent paper arXiv:1108.2387, we systematized inverse algorithm to obtain quiver gauge theory living on the M2-branes probing the singularities of special kind of Calabi-Yau four-folds which were complex cones over toric Fano ${\mathbb{P}}^3$, ${\mathcal{B}}_1$, ${\mathcal{B}}_2$, ${\mathcal{B}}_3$. These quiver gauge theories cannot be given a dimer tiling presentation. We use the method of partial resolution to show that the toric data of ${\mathbb{C}}^4$ and Fano ${\mathbb{P}}^3$ can be embedded inside the toric data of Fano $\mathcal{B}$ theories. This method indirectly justfies that the two node quiver Chern-Simons theories corresponding to ${\mathbb{C}}^4$, Fano ${\mathbb{P}}^3$ and their orbifolds can be obtained by higgsing matter fields of the three node parent quiver corresponding to Fano ${\mathcal{B}}_1$, ${\mathcal{B}}_2$, ${\mathcal{B}}_3$, ${\mathcal{B}}_4$ three-folds.

22 pages, 8 figures

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Cited by in corpus (1)

• Embedding and partial resolution of complex cones over Fano threefolds